ریسک غیر سیستماتیک و شکل گیری ائتلاف در بازارهای محصولات

We study the conjecture that increasing market volatility leads to larger coalitions in an oligopoly. Here, coalition formation decisions are made in a noncooperative game by risk averse firms. They use a sequential offer–counter-offer procedure initiated by Selten and Rubinstein. We find that the conjecture generally fails in a small oligopoly whose firms play a unanimity game, but it is validated in an oligopoly that allows open membership. However, it is valid in a small oligopoly if market volatility is sufficiently high, whatever the rule of membership.

We wish to study the relationship between unsystematic risk in the product market and coalition formation of oligopolistic firms. Specifically, we conjecture that increasing market volatility leads to larger coalitions. Starting from the assumption that firms are risk averse, firms attempt to avoid risk — which we identify with market volatility — by sharing it, and they do that by forming coalitions. Coalitions here are taken to be a subset of firms that coordinate their courses of action. Examples include complete mergers and cartels whose self-enforcing actions are coordinated. Our model is based on the noncooperative game that was initiated by Selten (1981) and Rubinstein (1982): players make sequential proposals to form coalitions. Their decision to participate in a coalition rests on an extensive form game of offers and counter offers. In another branch of the literature, externalities are permitted in such an extensive form game. Bloch (1995) studied associations in a two-stage model that yielded externalities to the participating firms which compete with each other and with outsiders in the market in the second stage. The sequential coalition formation decision is made in the first stage. In another paper, Bloch (1996) considers a more general sequential coalition model, assuming that the division of a coalition’s payoff is fixed. Brown and Chiang study the role of externalities in the form of cost saving in mergers and cartel formation (1997); they also examine managerial incentives and other externalities in mergers and cartels (1998). Conditions for a unique subgame perfect equilibrium of a sequential proposal game with externalities are given by Brown and Chiang (1999). Yi (1997), and Ray and Vohra (1997) make valuable contributions to this literature; they categorize various rules under which noncooperative sequential games proceed, analyze efficiency, stability, and similar properties of a sequential proposal model that involve externalities.1 In the externality literature of which this paper is a part, the payoff for each firm in a coalition depends on the coalition structure as a whole. To illustrate the coalition structure, suppose that, ex ante, there are three identical firms in an oligopoly. Label them sequentially. Four potential coalition structures can potentially form. One includes Firms 2 and 3 in a coalition and Firm 1 as a singleton; a second has all three firms are singletons; a third coalition structure includes Firms 1 and 2 in a coalition, while Firm 3 is a singleton; and the last is the grand coalition itself. (Symmetry obviates the need to consider the coalition structure in which Firms 1 and 3 coalesce, while Firm 2 remains a singleton.) Externalities can be expressed as shifts in the payoffs to all three firms in the four coalition structures for given output rates of the firms — for example, the payoff to Firm 1 in the first coalition structure may differ from the payoff to Firm 1 in the second coalition structure. The reason that externalities appear in the sequential coalition formation game is because firms are assumed to be farsighted and consider the consequences of their moves. As their turn to propose or respond to a coalition proposal comes up, they evaluate the effects of the various coalition structures on any coalition decision they make. This essential property is developed more fully in the literature on sequential coalition formation.2 Some studies have touched on the conjecture. Dewey (1961) claims that mergers are simply an efficient way to transfer assets from failing to successful firms; and there appears to be some empirical evidence for that hypothesis. If all firms are risk averse with declining risk aversion successful firms would tend to be less risk averse than failing firms and hence their acquisition can be interpreted as a risk spreading device. Gort (1969) argues that economic disturbances generate discrepancies in valuation of the type needed to produce mergers. One consequence is that the variance in valuations increases. It is a standard implication of risk aversion that the pooling of resources and the spreading of risk in an attempt to realize a rate of return that approaches the expected rate. We take this to be a natural motive for coalition formation. Of course, it is a consequence of the conjecture that firms and industries in relatively more risky endeavors — like those whose demand is more subject to cyclical factors than others in less volatile environments — are more likely to attempt to share risk and diversify, an empirical implication that we have not seen systematically tested. It is sufficient, both to validate and refute our conjecture, that we consider a simple, symmetric, two-stage, quantity-setting oligopoly model. In the first stage, firms make their self-enforcing coalition decisions, while in the second stage, they choose outputs as their strategies. All firms are identical, ex ante. Of course, in such a symmetric model the intra-coalitional share of the coalition’s payoff requires no cooperative game to determine it since all firms in the coalition receive the same payoff. Hence, the self-enforcing property is preserved in all aspects of the model. Though we focus on the symmetric model, some examples are given in which firms’ markets differ because of different random disturbances. These examples support the symmetric results and offer more suggestions for further inquiry. Central to our results is the rule by which coalitions can form. In a symmetric oligopoly firms can play an open membership game in which membership is open to all players who find it in their interest to adopt a joint output strategy. Alternatively, they can play by a unanimity rule in which a coalition forms if and only if all potential members agree to form it. It turns out the conjecture is sensitive to these rules. Our principal findings show the conjecture, increasing risk leads to larger coalitions, is not robust with regard to the rules of coalition formation. It fails, generally, in an oligopoly whose firms play by a unanimity rule but it is validated when they allow open membership. Interpretations and examples are given for these and other results that pertain to the inter-correlations of market volatility. Nevertheless, in spite of the different outcomes due to the two rules, we can say that if the volatility is sufficiently high, the conjecture holds irrespective of the rule by which coalitions form. Our model, which incorporates coalition formation with externalities and market volatility, is very simple. It is a complete information model in which unsystematic risk is represented by an additive random disturbance to demand. The model itself has an illustrious history, the demand specification first being used by Bowley (1924) and later by Spence (1976) and Dixit (1979). Products are taken to be imperfectly substitutable, with more differentiation raising the demand curves facing all firms. Thus, the implied market size increases as the degree of differentiation rises. This formulation is different from the other comparable formulation (see Shubik and Levitan, 1980). Modelling demand uncertainty has drawn considerable attention since Weitzman (1974). In their important contributions, Klemperer and Meyer, 1986 and Klemperer and Meyer, 1989 adopt the supply function approach and show that additive demand uncertainty can give rise to supply function equilibria. This endogenizes the firms’ decisions to set quantity as in the Cournot model, or price as in the Bertrand model. When products are differentiated, it turns out that the supply function equilibria resemble a Cournot or a Bertrand equilibrium, depending on the nature of technology. More specifically, when marginal costs slope upward (downward), the unique Nash equilibrium involves firms choosing quantities (prices). Since we assume that marginal cost is constant in our model, the following Klemperer and Meyer (1986), pp. 1269–1270) results are especially pertinent: ‘…with differentiated products, even in the limit as marginal cost becomes constant…, the equilibrium supply functions remain upward-sloping.’ In this framework, the quantity-setting-Cournot model is selected by firms. With differentiated products, firms possess some monopoly power in their own market and consequently, the supply function does not tend to be flat even when marginal costs are constant. Prices are above Bertrand levels. In short, the use of the quantity-setting, Cournot model and the assumption of constant marginal costs on which our analysis is based are not inconsistent. Needless to say, caution should be exercised when technology exhibits decreasing returns. In this case, the Bertrand model is more appropriate, so that whether the conjecture holds generally is an open question. Though there is a large literature that assumes uncertainty as we do, it does exact a price — namely, the assumption that firms are risk averse.3 This runs counter to a result from classical portfolio theory where it is known that the strategy of diversification — investing small fractions of wealth in each of a large number of noncorrelated securities — reduces risk. In the limit, any portfolio in which an investor’s wealth is invested equally in N securities with independently distributed returns achieves a riskless return as N increases beyond bound ( Werner, 1997). A risk averse investor would prefer the limit riskless return to the return of any portfolio. In effect, all investor risk is diversified away in equilibrium if all investors are small, there are perfect capital markets in which all securities are not perfectly correlated, and agency problems in firms are non-existent. A large number of studies in various fields — for example, contract, portfolio, and insurance theory — implicitly or otherwise justify the assumption that firms are risk neutral by appealing to the classical result that investors, qua owners, achieve a risk neutral allocation in equilibrium. This risk neutrality property rests on the assumption that investors are small. If for some reason N does not increase beyond bound all risk need not be diversified away, under-diversification is the result, at least for some investors. Thus the assumption of risk neutrality on the part of all investors in equilibrium is not valid. In particular, a branch of the portfolio literature studies concentrated ownership, where it is found that the ownership structure of firms affects their payoffs by influencing the amount of monitoring of management ( Admati et al., 1994, Zhang, 1998 and Black, 1992). 4 Firms then are subject not only to systematic risk but also to unsystematic risk. The results concerning coalition formation and unsystematic risk in this paper apply to such firms. Managerial ownership has also been adduced as a reason that those firms are risk averse. Because managers have a personal portfolio that is underdiversified after their commitments of human and financial capital, it is argued (Treynor and Black, 1976) that management’s risk aversion results in efforts to reduce risk. Empirical evidence consistent with this has been found by Amihud and Lev (1981), May (1995) and Chen et al. (1998). Another segment of the literature leads to the conclusion that concentrated ownership should not be unexpected. The general equilibrium literature under uncertainty (Arrow and Hahn, 1971) — assuming complete information and no oligopoly problems — justifies the canonical assumption of risk neutrality on the part of firms as follows. In this uncertainty model, the commodities consumed and the outputs produced by firms are defined not only by their physical properties but also by an event conditional upon when it is available, the states of the world. In a two period world of uncertainty, the true state of the world is unknown in the first period but it is revealed in the second. Under standard convexity assumptions, a competitive equilibrium results that yields a set of prices such that each price is associated with a claim on producers. The producers promise to supply a unit of a commodity if a certain state occurs and nothing otherwise. If there is a price for each good in each state, the firm need only maximize its profits, acting as if it were risk neutral. All the risk bearing is done by consumers as their risk preferences allow. That means that consumers effectively write and sell insurance policies. The result is that a competitive equilibrium exists in a world of uncertainty and all firms are risk neutral, though consumers need not be. However, change the model to allow for stock ownership as follows: retain the complete information assumption, and allow for a mixture of insurance markets, markets of shares of ownership in firms, and markets for other assets. Then, even the general equilibrium model produces a potential mismatching of owners’ preferences and the production possibilities of firms (Dreze, 1974). In a two-firm, two-state, two-consumer example, Dreze allows each risk neutral firm to produce optimally, given the consumption preferences of its owner. With the firms’ production plans taken as given, and with shares in both firms selling at the same price, he shows that the optimal portfolios and the production plans are inefficient. One way to achieve efficiency is to have major changes in the composition of ownership. As Dreze explains: ‘The initiative now lies with the consumers. But this requires additional information on their part: each consumer must know the production set of the other firm, not only its production plan. Furthermore, major changes in ownership fractions are again required: in order to exert enough influence on the decision within the firm to move its production plan (in the appropriate direction), the consumer may have to acquire a majority interest.’ (Dreze, 1974, pp. 149–150). In short, the introduction of uncertainty into a general equilibrium model in which shares of firms are owned by consumers can result in a mismatching inefficiency; and hence, there is an incentive to concentrate ownership of firms in order to bring production plans more in line with consumer preferences. Summing up, concentrated ownership can arise from the incentive to monitor management in order to avoid agency problems or to align production plans with owners’ preferences. In either event, ownership can be expected to be concentrated, leading to under-diversification of owners’ financial portfolio and, when firms are managed by owners, under-diversification, also, of human capital endeavors. For this reason, firms themselves attempt to reduce market risk. The paper is organized as follows. The assumptions and notation of the Cournot-Nash model under uncertainty immediately follows. The second stage output decisions, along with certain results involving coalitional output and market volatility, are then presented. This is followed by the first stage analysis of sequential coalition formation and our principal findings. Concluding remarks complete the paper.

Our analysis applies to quantity-setting oligopolistic firms that are risk averse. Accepting this, our conjecture has considerable intuitive appeal. The desire to reduce risk by sharing it is a natural consequence of risk aversion, and coalition formation is a natural way to share risk. Hence, there is considerable force to the conjecture that more market volatility leads to larger coalitions. But the conjecture is not vacuous because it does not hold in all cases. A change in the rule of how a coalition forms, from open membership to a unanimity rule, can invalidate it. Our simple symmetric model suffices both to confirm the conjecture in the open membership rule case and to refute it when unanimity is required for a coalition to form. Though the conjecture does not hold, in general when the unanimity rule applies, neither does it always fail. In fact, the conjecture holds nicely for higher ranges of the market volatility measure. If the variance of demand increases from a level in a middle range to a higher level, coalition size increases, no matter what the rule. It only fails when the market volatility is sufficiently low and proceeds to a middle range. Considerable volatility is required to overcome coalition-destroying factors in unanimity based coalitions — the principal factor being the Stigler effect, which is the desire by firms to remain outside coalitions in order to free ride on the price rise effected by the increase in the monopoly power of a coalition. In sum, if market volatility starts at a low level and proceeds to a moderate level, do not expect a general increase in coalition size. But if it is sufficiently large, coalition size rises with market volatility, independently of the coalition-forming rule. Our analysis rests on a complete information model and the assumption that uncertainty resides in the demand for the products. Whether the results hold for asymmetric information concerning demand uncertainty or cost uncertainty, this relaxation of the complete information assumption in its several directions remains to be pursued. Also, the paper raises immediate caveats concerning the assumptions that firms compete in quantities and that marginal costs are constant. Clearly, these have to be relaxed. It remains an open question whether our results can be extended to the decreasing marginal cost case, and especially to the subcase where only a Bertrand equilibrium can be obtained. We add that with regard to the latter, there is a need to examine coalition formation within Bertrand price-setting oligopolies. These are worthy subjects for future research.